Before thinking about abolishing private school you should look at what actually is the difference between state schools and private schools. I go to a private school (my parents are literally spending their last penny to send me there so they are not wealthy and my not arrogant or posh, just a normal guy (i hope :P )) but I have also been to a state school. I was talking to a teacher at my school about this. She has worked in both state schools and private schools. Now look at just her, her teaching didn't improve from being in a private school. She also said that the majority of pupils in the state school were no different from most of us in the private. We we all kind, hardworking, got along, were quite clever.
The differences were this: Smaller class sizes in private schools (I'm in A level and my class sizes and 3, 4 and 5 and the school has 250 pupils, which is less than some years in state schools). More disruptive pupils in a class (due to bigger class sizes and people who don't want school). At Private schools, if you mess around a lot, bully other pupils etc. you are punished very quickly and harshly in cases (fair enough, bullying should not be tolerated) but at state schools the punishments aren't as effective (my teachers words).
The differences in the standard of teaching aren’t that great. At my school I have had some absolutely awful teachers, no one in the class knew anything about the subject lots had to get tutors (myself included) and then we did well and equally there are some amazing teachers at state schools.
Anyway, of course this is very generalised and taken from the experience of a few people but basically my point is that we should look at what are the real differences in the experiences children have at state schools and private schools and look to reduce those changes in the state system to help, e.g. perhaps reducing class size in state schools (obviously would be harder in practice) but something like that could help.
This is just my opinion but essentially, no we shouldn't ban them